Assuming that the Batsignal means that Batman is officially deputized by Gotham and the GCPD, he couldn’t morally kill the Joker intentionally accept in circumstances in which an officer would be morally sanctioned to use lethal force.

If he stopped the Joker using non-lethal force and then simply decided to kill him “because he’s bad,” he would be morally guilty of murder, even if he were never convicted of it in a court of law.

Morally speaking, no private person ever has the moral authority to intentionally take a life, though one may use force which could be lethal if it is the only way to defend oneself and others.

Those imputed with the authority to defend the common good may use lethal force only when this is sanctioned by the law to defend the common good against aggressors.

This is the teaching of St. Thomas Aquinas and of the Church as I understand it.